What is the velocity of the COM of a falling rod at a given angle?

In summary, the conversation discusses determining the velocity of the center of mass when a rod falls from an upright position at an angle θ with the horizontal. The solution involves using conservation of angular momentum and the work energy theorem, but it is noted that there may be a mistake as the angular momentum is not conserved due to an external torque on the rod.
  • #1
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Homework Statement


Given a rod of length 'l' placed upright on a smooth floor.A slight disturbance causes it to fall.Now required is the velocity with which the COM falls down when the rod makes angle θ with horizontal.

The Attempt at a Solution


now let the middle portion of the rod be A,and the lower portion B.Using conservation of angular momentum,we get
0=(ml^2/12)ω-m*Va*l/2*cosθ
By work energy theorem,
mgl/2*(1-sinθ)=m(Va)^2/2+ml^2/12*ω^2 (ω about point B)
What is the mistake I'm making here?
 
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  • #2
Angular momentum is not conserved. The table has a torque on the COM of the rod, meaning there is an external force which has a torque.
 

FAQ: What is the velocity of the COM of a falling rod at a given angle?

What is rotational dynamics?

Rotational dynamics is the study of the motion of objects that are rotating around an axis. It involves understanding the relationship between angular velocity, angular acceleration, torque, and moment of inertia.

Why is rotational dynamics important?

Rotational dynamics is important because many natural phenomena, such as the rotation of the Earth, the motion of planets, and the movement of molecules, can be described using rotational dynamics principles. It also has practical applications in engineering, such as designing machines and structures that rotate.

What is the difference between linear and rotational dynamics?

The main difference between linear and rotational dynamics is the type of motion being studied. Linear dynamics deals with the motion of objects in a straight line, while rotational dynamics deals with the motion of objects around an axis. Additionally, rotational dynamics involves concepts such as moment of inertia and torque, which are not present in linear dynamics.

How is angular momentum related to rotational dynamics?

Angular momentum is a key concept in rotational dynamics. It is a measure of an object's tendency to continue rotating around an axis, and is the product of its moment of inertia and angular velocity. In other words, the greater the angular momentum, the harder it is to stop an object's rotation.

What are some real-life examples of rotational dynamics?

Some examples of rotational dynamics in everyday life include the spinning of a top, the rotation of a bicycle wheel, and the swinging of a pendulum. In sports, rotational dynamics can be seen in the spin of a basketball or the rotation of a figure skater. In engineering, it is used in designing car engines, wind turbines, and other rotating machinery.

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